Von Staudt–Clausen theorem
Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B2n for every prime p such that p − 1 divides 2n, we obtain an integer, i.e.,
This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B2n as the product of all primes p such that p − 1 divides 2n; consequently the denominators are square-free and divisible by 6.
These denominators are
- 6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... (sequence A002445 in OEIS)
A proof of the Von Staudt–Clausen theorem follows from an explicit formula for Bernoulli numbers which is:
and as a corollary:
where are the Stirling numbers of the second kind.
Furthermore the following lemmas are needed:
Let p be a prime number then,
1. If p-1 divides 2n then,
2. If p-1 does not divide 2n then,
Proof of (1) and (2): One has from Fermat's little theorem,
If p-1 divides 2n then one has,
Thereafter one has,
from which (1) follows immediately.
If p-1 does not divide 2n then after Fermat's theorem one has,
If one lets (Greatest integer function) then after iteration one has,
for and .
Thereafter one has,
Lemma (2) now follows from the above and the fact that S(n,j)=0 for j>n.
(3). It is easy to deduce that for a>2 and b>2, ab divides (ab-1)!.
(4). Stirling numbers of second kind are integers.
Proof of the theorem: Now we are ready to prove Von-Staudt Clausen theorem,
If j+1 is composite and j>3 then from (3), j+1 divides j!.
If j+1 is prime then we use (1) and (2) and if j+1 is composite then we use (3) and (4) to deduce:
- H. Rademacher, Analytic Number Theory, Springer-Verlag, New York, 1973.
- T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 197.
- Clausen, Thomas (1840), Theorem, Astronomische Nachrichten 17 (22): 351–352, doi:10.1002/asna.18400172204
- Rado, R. (1934), A New Proof of a Theorem of V. Staudt, J. London Math. Soc. 9 (2): 85–88, doi:10.1112/jlms/s1-9.2.85
- von Staudt, Ch. (1840), Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend, Journal für Reine und Angewandte Mathematik 21: 372–374, ISSN 0075-4102, JFM 021.0672cj